## Line Chart: Number of LCPs vs N-labels with Inference Time Comparison ### Overview The chart compares three metrics across N-labels (20-60): 1. Number of LC-Prototypes (orange line) 2. Original inference time/query (dashed blue line) 3. Optimized inference time/query (solid blue line) All axes use logarithmic scales to emphasize exponential trends. ### Components/Axes - **X-axis**: N-labels (20, 30, 40, 50, 60) - **Left Y-axis**: # LC-Prototypes (log scale: 10⁰ to 10⁵) - **Right Y-axis**: Inference time (ms/query) (log scale: 10⁰ to 10⁵) - **Legend**: - Orange circles: Number of LCPs - Dashed blue: Original inference time - Solid blue: Optimized inference time ### Detailed Analysis 1. **Number of LCPs (Orange Line)** - 20 N-labels: 487 prototypes - 30 N-labels: 7,853 prototypes - 60 N-labels: 53,640 prototypes - *Trend*: Exponential growth (doubles between 20→30, triples between 30→60) 2. **Original Inference Time (Dashed Blue Line)** - 20 N-labels: 21ms/query - 30 N-labels: 306ms/query - 60 N-labels: 2,170ms/query - *Trend*: Exponential increase (14x between 20→30, 7x between 30→60) 3. **Optimized Inference Time (Solid Blue Line)** - 20 N-labels: 2ms/query - 30 N-labels: 4ms/query - 60 N-labels: 17ms/query - *Trend*: Linear growth (2x between 20→30, 4.25x between 30→60) ### Key Observations - LC-Prototype count grows **exponentially** with N-labels (16x increase from 20→60) - Original inference time grows **exponentially** (100x increase from 20→60) - Optimized inference time grows **linearly** (8.5x increase from 20→60) - At 60 N-labels, optimized time is **128x faster** than original (2,170ms vs 17ms) ### Interpretation The chart demonstrates a critical tradeoff: 1. **LC-Prototype Explosion**: The number of prototypes grows exponentially with N-labels, suggesting combinatorial complexity in label combinations. 2. **Optimization Impact**: The optimized inference time shows near-linear scaling despite increasing N-labels, indicating algorithmic improvements (e.g., caching, parallelization, or model distillation). 3. **Practical Implications**: - At 60 N-labels, original systems would take ~2 seconds per query vs optimized 17ms – a 128x speedup. - The log scales reveal that raw LC-Prototype counts (10⁴-10⁵ range) are impractical for real-time systems without optimization. - The optimized line's near-linear trend suggests the system avoids combinatorial explosion through smart resource allocation or approximation techniques. *Note: All values are extracted from labeled data points; intermediate values (e.g., 40 N-labels) are interpolated from the logarithmic trend.*